New Gravitational Memories
Sabrina Pasterski, Andrew Strominger and Alexander Zhiboedov
Center for the Fundamental Laws of Nature
Harvard University, Cambridge, MA 02138 USA
Abstract
The conventional gravitational memory effect is a relative displacement in the position
of two detectors induced by radiative energy flux. We find a new type of gravitational
‘spin memory’ in which beams on clockwise and counterclockwise orbits acquire a relative
delay induced by radiative angular momentum flux. It has recently been shown that the
displacement memory formula is a Fourier transform in time of Weinberg’s soft graviton
theorem. Here we see that the spin memory formula is a Fourier transform in time of the
recently-discovered subleading soft graviton theorem.
Contents
1. Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Asymptotically flat metrics
1
. . . . . . . . . . . . . . . . . . . . . . . .
3
3. Displacement memory effect . . . . . . . . . . . . . . . . . . . . . . . .
6
4. Spin memory effect
6
. . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Spin memory and angular momentum flux
6. Equivalence to subleading soft theorem
7. An infinity of conserved charges
. . . . . . . . . . . . . . . . .
8
. . . . . . . . . . . . . . . . . . 10
. . . . . . . . . . . . . . . . . . . . . . 12
Appendix A. Massless particle stress-energy tensor
. . . . . . . . . . . . . . . 15
1. Introduction
The passage of gravitational radiation past a pair of nearby inertial detectors produces
oscillations in their relative positions. After the waves have passed, and spacetime locally
reverts to the vacuum, the detectors in general do not return to their initial relative positions. The resulting displacement, discovered in 1974 [1-11], is known as the gravitational
memory effect. Direct measurement of the gravitational memory may be possible in the
coming years, see e.g. [12,13]. The effect is a consequence [14] of the fact that the radiation
induces transitions among the many BMS-degenerate [15] vacua in general relativity. The
initial and final spacetime geometries, although both flat, differ by a BMS supertranslation. The displacement is proportional to the BMS-induced shift in the spacetime metric,
which in turn is given by a universal formula involving moments of the asymptotic energy
flux.
Since the initial and final metrics differ, the Fourier transform in time must have a
pole at zero energy. A universal formula for this pole was found in 1965 [16] and is known
as Weinberg’s soft graviton theorem. The complete equivalence of the soft graviton and
displacement memory formulae was demonstrated in [14].
Recently, a new universal soft graviton formula was discovered [17](see also [18,19])
that governs not the pole but the finite piece in the expansion of soft graviton scattering
1
about zero energy. This was shown [20] to be a consequence of the BMS superrotations1 of
[21] in the same sense that Weinberg’s pole formula is a consequence of BMS supertranslations.
This discovery immediately raises the question: is there a new kind of gravitational
memory associated with superrotations and the subleading soft theorem? In this paper
we show that the answer is yes. While displacement memory is sourced by moments of
the energy flux through null infinity (I), the new memory is sourced by moments of the
angular momentum flux. Accordingly we call it spin memory. The spin memory effect
provides a cogent operational meaning to the superrotational symmetry of gravitational
scattering.
Spin memory has a chiral structure and cannot be measured by inertial detectors.
Instead, we consider light rays which repeatedly orbit (with the help of fiber optics or
mirrors) clockwise or counterclockwise around a fixed circle in the asymptotic region. The
passage of angular-momentum-carrying radiation will induce a relative time delay between
the counter-orbiting light rays, resulting for example in a shift in the interference fringe.
A universal formula for this delay is given in terms of moments of the angular momentum
flux through infinity. The relative time delay for counter-orbiting light rays is the spin
memory effect. It is a new kind of gravitational memory.
This paper is organized as follows. Section 2 outlines the metric and constraint equations for asymptotically flat spacetimes. Section 3 reviews the displacement memory effect.
Section 4 introduces the spin memory effect. Section 5 uses the constraint equations and
boundary conditions to relate this new memory effect to angular momentum flux. Section
6 demonstrates that the spin memory formula is the Fourier transform in time of the subleading soft graviton formula of [17]. Finally, in section 7, we discuss the infinite family
of conserved charges associated to the infinite superrotational symmetries. Measurements
verifying the conservation laws are described. We close with a short comment on implications for black hole information. The appendix derives several formulae concerning the
angular momentum of spinning particles on null geodesics.
1
The associated symmetry group is the familiar Virasoro symmetry of euclidean two-
dimensional conformal field theory [21]. This may be usefully embedded in a larger group of
all diffeomorphisms of the sphere [22].
2
2. Asymptotically flat metrics
The expansion of an asymptotically flat spacetime metric near I + in retarded Bondi
coordinates takes the form2
mB 2
du
r
1 4
1
+ rCzz dz 2 + Dz Czz dudz + ( Nz − ∂z (Czz C zz ))dudz + c.c. + ...
r 3
4
ds2 = −du2 − 2dudr + 2r2 γzz̄ dzdz̄ + 2
where u = t − r, γzz̄ =
2
(1+z z̄)2
(2.1)
is the unit metric on S 2 (used to raise and lower z and z̄
indices), Dz is the γ-covariant derivative, and subleading terms are suppressed by powers
of r. The Bondi mass aspect mB , the angular momentum aspect Nz , and Czz are functions
M
of (u, z, z̄), not r. They are related by the I + constraint equations Guu = 8πGTuu
1 2 zz
Dz N + Dz̄2 N z̄z̄ − Tuu ,
4
1
M
≡ Nzz N zz + 4πG lim [r2 Tuu
],
r→∞
4
(2.2)
1 2 zz
∂z Dz C − Dz̄2 C z̄z̄ + ∂z mB − Tuz ,
4
1
1
M
≡ 8πG lim [r2 Tuz
] − Dz [Czz N zz ] − Czz Dz N zz ,
r→∞
4
2
(2.3)
∂u mB =
Tuu
M
and Guz = 8πGTuz
∂u Nz =
Tuz
where Nzz = ∂u Czz is the Bondi news, T M is the matter stress tensor and Tuu (Tuz ) is
the total energy (angular momentum) flux through a given point on I + . The angular
momentum aspect is related to the Weyl tensor component Ψ01 on I + by
Nz = lim r3 Czrru .
(2.4)
1
1
ImΨ02 = Im lim rγ zz̄ Cuz̄zr = −Im[ Dz2 C zz + Czz N zz ].
r→∞
2
4
(2.5)
r→∞
We also note that
Most of our discussion will concern I + , but the metric expansion near I − in the
retarded Bondi coordinate v = t + r is
mB 2
dv
r
1 4
1
+ −rCzz dz 2 + Dz Czz dvdz + ( Nz + ∂z (Czz C zz ))dvdz + c.c. + ...
r 3
4
ds2 = −dv 2 + 2dvdr + 2r2 γzz̄ dzdz̄ + 2
2
(2.6)
Our definition of Nz , which is proportional to the Weyl tensor (see below) is related to NzBT
of [21] by 4Nz = 4NzBT + Czz Dz C zz + 34 ∂z (Czz C zz ).
3
where here the metric perturbations are functions of (v, z, z̄). The z coordinate on I − is
defined so that (−v, z, z̄) is the P T conjugate of (u, z, z̄): hence they lie on the same null
generator of I and are antipodally located relative to the origin of the spacetime. The I −
M
constraint equations become Gvv = 8πGTvv
∂v mB = −
Tvv ≡
1 2 zz
Dz N + Dz̄2 N z̄z̄ + Tvv ,
4
1
M
Nzz N zz + 4πG lim [r2 Tvv
],
r→∞
4
(2.7)
M
and Gvz = 8πGTvz
1 ∂v Nz = − ∂z Dz2 C zz − Dz̄2 C z̄z̄ + ∂z mB + Tvz ,
4
1
1
M
Tvz ≡ 8πG lim [r2 Tvz
] − Dz [Czz N zz ] − Czz Dz N zz .
r→∞
4
2
(2.8)
+
+
We use the symbol I−
(I+
) to denote the past and future S 2 boundaries of I + , and
−
I±
for those of I − . In this paper, we consider spacetimes which decay to the vacuum in
+
−
. (The more general case requires an analysis of extra
and I+
the far past and future I−
past and future boundary terms.) In particular, we require
Nz |I + = Nz |I − = mB |I + = mB |I − = 0.
−
+
−
+
(2.9)
±
Moreover near all the boundaries I±
of I, the geometry is in the vacuum in the sense that
the radiative modes are unexcited:
Nzz |I ± = ImΨ02 |I ± = 0.
±
±
(2.10)
More precisely, following Christodoulou and Klainerman [23], we take Nzz ∼ |u|−3/2 (|v|−3/2 )
on I + (I − ) as well as a corresponding falloff of the stress tensor.
These conditions do not imply Czz |I ± = 0. Rather, the general solution of (2.10) is
±
(see e.g [24])
Czz = −2Dz2 C,
(2.11)
where C is any (u-independent) function of (z, z̄). These solutions are mapped to one
another by BMS supertranslations and exhibit the large vacuum degeneracy in general
relativity.
4
As described in [24], to define gravitational scattering one must specify boundary or
continuity conditions on mB and Czz near where I + and I − meet. The unique Lorentz,
PT and BMS-invariant choice is simply
Czz |I + = Czz |I − ,
−
mB |I + = mB |I − .
−
+
(2.12)
+
In this paper Nz plays an important role and its determination from the constraint equation
(2.3) also requires a continuity condition. This is a bit tricky because outside the center±
of-mass frame, Nz may grow linearly with u or v near I±
. It follows from (2.3) and (2.10)
that the divergent term is exact: Nz ∼ u∂z mB . Fortunately for us, such exact terms are
irrelevant for our purposes (see section 5). We will need a continuity condition for the curl
of Nz . (2.10), (2.12) and the Bianchi identity imply
∂[z Nz̄] |I − = ∂[z Nz̄] |I + .
(2.13)
−
+
We are interested in the difference between the initial and final functions C in (2.11)
on I. This can be determined by integrating the constraint (2.2) as follows (see [14] for
more details). Defining
∆+ mB = mB |I + − mB |I + = −mB |I + ,
∆+ Czz = Czz |I + − Czz |I + ,
+
−
+
−
−
(2.14)
and using (2.2) one finds
Dz2 ∆+ C zz
Z
=2
du Tuu + 2∆+ mB .
The ∆+ C which produces such a ∆+ Czz is obtained by inverting Dz2 Dz̄2 :
Z
Z
+
2
∆ C(z, z̄) = d wγww̄ G(z; w)
du Tuu (w) + ∆mB
(2.15)
(2.16)
where the Green’s function is given by
G(z; w) = −
1
Θ
Θ
sin2 log sin2 ,
π
2
2
sin2
Θ(z, w)
|z − w|2
≡
.
2
(1 + ww̄)(1 + z z̄)
(2.17)
An equation similar to (2.16) may be derived for the shift of C on I − . Adding the
two equations, using the boundary condition (2.12), and defining
∆C = ∆+ C − ∆− C,
one arrives at the simple relation
Z
Z
Z
2
∆C(z, z̄) = d wγww̄ G(z; w)
du Tuu (w) − dv Tvv (w) .
5
(2.18)
(2.19)
3. Displacement memory effect
In this section, we briefly review the standard gravitational memory effect. The passage of a finite pulse of radiation or other form of energy through a region of spacetime
produces a gravitational field which moves inertial detectors. The final positions of a pair
of nearby detectors are generically displaced relative to the initial ones according to a
simple and universal formula [1-11], which we now review briefly.
Consider two nearby inertial detectors with proper worldline tangent vectors tµ and
relative displacement vector sµ . We take the worldlines to be at large r and extend for
infinite retarded time near I + . sµ evolves according to the geodesic deviation equation
∂τ2 sµ = Rµλρν tλ tρ sν ,
(3.1)
where τ is the detector’s proper time. At large r in the geometry (2.1), we may approximate
τ ∼ u, tλ ∂λ = ∂u and
1
Rzuzu = − r∂u2 Czz .
2
(3.2)
γ zz̄ 2
∂ Czz sz .
2r u
(3.3)
It follows that
∂u2 sz̄ =
Integrating twice one finds, to leading order in 1r , a net change in the displacement
∆+ sz̄ =
γ zz̄ +
∆ Czz sz ,
2r
(3.4)
where ∆+ Czz is given in term of moments of the asymptotic energy flux by the second
derivative of (2.16). (3.4) is the standard displacement memory formula.
4. Spin memory effect
This section describes the new spin memory effect, which affects orbiting objects such
as protons in the LHC, or signals exchanged by eLISA detectors. Consider a circle C of
radius L near I + centered around a point z0 on a sphere of large fixed r = r0 , where
L << r0 . This is described by
2
Leiφ 1 + z0 z̄0
L
√
+O
Z(φ) = z0 1 +
2r0
z0 z̄0
r02
6
(4.1)
where φ ∼ φ + 2π. A light ray in either a clockwise or counterclockwise orbit (aided by
mirrors or fiber optics) along C starting at φ(0) = 0, has a trajectory φ(u) that obeys
ds2 = 0
= 1 − 2r02 γzz̄ ∂u Z∂u Z̄ − 2
mB
− r0 Czz (∂u Z)2 − r0 Cz̄z̄ (∂u Z̄)2
r0
(4.2)
− [Dz Czz ∂u Z + Dz̄ Cz̄z̄ ∂u Z̄] + ...
where it is here and hereafter assumed that Czz does not change significantly over a single
period. To this order, only the term in square brackets in (4.2) is odd under ∂u Z → −∂u Z.
If two light rays are simultaneously set in orbit in opposite directions, the times at which
they return to φ = 03 will differ by the u-integral of this odd term
I
∆P =
Dz Czz dz + Dz̄ Cz̄z̄ dz̄ .
(4.3)
C
This formula in fact applies to any contour C, circular or not.
At first glance, it appears from (4.3) that the returns of the counter-orbiting light rays
are desynchronized, even in the vacuum, as long as Czz is nonzero. In fact this is not the
case. For Czz of the vacuum form (2.11), one readily finds that
I
∆Pvacuum = −2
d(Dz Dz C + C) = 0.
(4.4)
C
Hence desynchronization occurs only during the passage of radiation through I + . The
total relative time delay, integrated over all orbits is
1
∆ u=
2πL
+
Z
I
du
Dz Czz dz + Dz̄ Cz̄z̄ dz̄ .
(4.5)
C
This leads to a shift in the interference pattern between counter-orbiting light pulses. The
shift is an infrared effect proportional to the u-zero mode of Czz . This is the spin memory
effect.
3
The line φ = 0 is a geodesic in the induced geometry of the ring (4.2) only in the limit
r → ∞. Detectors at fixed φ are ‘BMS detectors’ of the type discussed in [14]. A finite-r geodesic
detector will be boosted and observe a different ∆P .
7
5. Spin memory and angular momentum flux
Displacement memory (3.4) can be expressed as an integral of the net local asymptotic
energy flux convoluted with the Green’s function (2.19) on the sphere. In this section we
derive an analogous formulae for spin memory as a convoluted integral involving the net
local asymptotic angular momentum flux.
Taking ∂z̄ of the Guz constraint in (2.3) and ∂z of the complex conjugate Guz̄ constraint gives:
∂z ∂z̄ mB = Re[∂u ∂z̄ Nz + ∂z̄ Tuz ]
(5.1)
Im[∂z̄ Dz3 C zz ] = 2Im[∂u ∂z̄ Nz + ∂z̄ Tuz ].
(5.2)
Multiplying (5.2) by the Green’s function
Θ
,
G(z; w) = log sin
2
2
|z − w|2
Θ(z, w)
sin
≡
2
(1 + ww̄)(1 + z z̄)
2
(5.3)
which obeys
1
∂z ∂z̄ G(z; w) = 2πδ 2 (z − w) − γzz̄ ,
2
(5.4)
and integrating over d2 z gives:
2 ww
πIm[Dw
C ]
Z
= −Im
d2 z∂z̄ G(z; w)[∂u Nz + Tuz ].
(5.5)
Note that the right hand side of (5.5) is invariant under shifts Nz → Nz + ∂z X for any
real X, so only the curl part of Nz contributes. Integrating both sides over the disk DC
whose boundary is C and using Stokes’ theorem, (5.5) leads to
I
Z
Z
w
w̄
2
π (D Cww dw + D Cw̄w̄ dw̄) = −2Im
d wγww̄ d2 z∂z̄ G(z; w)[∂u Nz + Tuz ]. (5.6)
C
DC
Multiplying by
1
2π 2 L
and integrating over u then yields
1
∆ u = − 2 Im
π L
+
Z
2
d wγww̄
Z
Z
+
d z∂z̄ G(z; w) ∆ Nz + duTuz .
2
(5.7)
DC
where ∆+ Nz ≡ Nz |I + − Nz |I + = −Nz |I + is the shift in the angular momentum as+
−
−
pect. In many applications – for example geometries which are initially asymptotically
Schwarzschild through subleading order – this term will vanish. Moreover, if ∆+ Nz is
exact, i.e. ∆+ Nz = ∂z X for any real X, no contribution to the imaginary part appears in
(5.7). Hence (5.7) depends only on the curl of ∆+ Nz .
8
A similar analysis near I − leads to the formula
1
∆ v = − 2 Im
π L
−
Z
Z
2
d wγww̄
Z
−
d z∂z̄ G(z; w) −∆ Nz + dvTvz .
2
(5.8)
DC
where the contour C and disk DC on I − is defined by the curve defined in equation (4.1)
on I − . This means it will lie in the antipodal spatial direction from the origin. Using the
continuity condition (2.13) on Nz one finds, in analogy to (2.19), an expression relating
time delays and fluxes
1
∆τ ≡ ∆ u − ∆ v = − 2 Im
π L
+
−
Z
Z
2
2
Z
d z∂z̄ G(z; w)
d wγww̄
Z
duTuz −
dvTvz . (5.9)
DC
The right hand side of (5.9) is related to the local angular momentum flux through I.
Consider the case when Tuz arises from massless particles or localized wave packets which
puncture I + at points (uk , zk ). Then, as we show in the appendix,
Tuz = 8πG
X
k
2
δ (z − zk )
i
,
δ(u − uk ) Luz (zk ) − hk ∂z
2
γzz̄
(5.10)
together with a similar formula for Tvz . Here Luz (zk ), (hk ) is the orbital (spin) angular
momentum of the kth particle associated to a rotation around (boost towards) the point
zk on the sphere. The leading contribution from such particles to the time delay is
Z
8G X zk z̄k
γ
∆τ = −
Im
d2 wγww̄ Luz (zk )∂z̄k G(zk ; w) + πhk∈C .
(5.11)
πL
DC
k
The second term in (5.11) has the simple interpretation that an object of spin hk passing
through C at (or near) lightspeed induces a time delay of order
hk
L ,
with no factors of r0 .
This can be understood as the frame-dragging effect. If C lies a distance of order L from
zk , the first term is typically a factor of
L
r0
smaller than the second.
Another interesting case is outgoing quadrupole radiation, with no incoming news or
Tvz on I − .4 The displacement memory effect for configurations of this type is of potential
astrophysical interest and was analyzed in [6]. This is described by the news tensor on I +
Nzz = N Yzi Yzj
4
(5.12)
Since we are taking mB |I − = 0 here, the initial energy would have to enter in a spherically
−
−
symmetric wave from I .
9
for some N (u). As an example, take i = j, Yiz = z and
N=
2
α
∂u e−u +iωu .
1/4
(2π)
(5.13)
The resulting angular momentum flux obeys
Z
duTuz =
so that
Z
Im
Z
du
z 2 z̄ 2
3i 2
α ω∂z
,
2
(1 + z z̄)4
1
w2 w̄2
= 3πα ω
−
,
30 (1 + ww̄)4
2
2
d z∂z̄ G(z; w)Tuz
(5.14)
(5.15)
using (5.4). The quadrupole contribution to the time delay around a contour C becomes
3α2 ω
∆ u=
πL
+
Z
2
d wγww̄
DC
w2 w̄2
1
.
−
(1 + ww̄)4
30
For typical choices of C such that the area of DC is order L2 /r02 , we have ∆+ u ∼
(5.16)
α2 L
.
r02
6. Equivalence to subleading soft theorem
In the sixties, Weinberg [16] showed that scattering amplitudes in any theory with
gravity exhibit universal poles as the energy ω of any external graviton is taken to zero.
Recently [17,18,19,20,22,25,26] it has been shown that the finite, subleading term in the
ω → 0 expansion also exhibits universal behavior. The coefficient of the leading pole
was shown in [14] to be related by a timeline Fourier transform of the expression for
displacement memory. We now show that the subleading term is a Fourier transform of
the expression for spin memory.
The subleading soft graviton theorem is a universal relation between (n + 1)-particle
(with one soft graviton) and n-particle tree-level quantum scattering amplitudes [17]
1
(1) µν
( lim + lim )An+1 p1 , ...pn ; (ωq, µν ) = Sµν
An p1 , ...pn ,
2 ω→0+ ω→0−
where
(6.1)
n
(1)
Sµν
iκ X pk(µ Jkν)λ q λ
=
2
q · pk
(6.2)
k=1
and κ =
√
32πG. In this expression, the parentheses denote (µ, ν) symmetrization, q =
(ω, ω q̂) with q̂ 2 = 1 is the four-momentum, and µν is the transverse-traceless polarization
10
tensor of the graviton. We define incoming particles to have negative p0 and take ω positive
for an outgoing graviton. µ, ν indices refer to asymptotically Minkowskian coordinates
given in terms of retarded coordinates as
x0 = u + r,
2rz
,
x1 + ix2 =
1 + z z̄
r(1 − z z̄)
x3 =
,
1 + z z̄
(6.3)
or in terms of advance coordinates as
x0 = v − r,
2rz
,
x1 + ix2 = −
1 + z z̄
r(1 − z z̄)
x3 = −
.
1 + z z̄
(6.4)
The symmetrized limit in (6.1) projects out the leading Weinberg pole, leaving the subleading finite term of interest here. The linearized expectation value of the asymptotic
metric fluctuation produced in the n-particle scattering process obeys the semiclassical
momentum space formula
An+1 p1 , ...pn ; (ωq, µν )
( lim + lim )hαβ (ω, q) = αβ ( lim + lim )
ω→0+
ω→0−
ω→0+
ω→0−
An p1 , ...pn
n
X
pkµ Jkνλ q λ
= iκαβ µν
.
q · pk
(6.5)
k=1
In the last line, and in similar expressions below, an expectation value of the expression
involving the differential operator Jkνλ acting in the matrix element in An is implicit.
Expression (6.5) characterizes linearized fields by their momenta whereas the new
memory formula (5.9) is given in terms of I values of fields. These are simply related.
Using the large-r stationary phase approximation as in [20,27]
Z
Z
duCzz (u, q̂) −
xµ
r→∞ r
where X µ ≡ lim
dvCzz (v, q̂) = −( lim + lim )
ω→0+
ω→0−
iκ
∂z X µ ∂z X ν hµν (ω, q̂),
8π
(6.6)
and the unit vector q̂ is viewed as a coordinate on I. The hard particle
momenta pk , the soft graviton momentum q, and complex polarization −µν = −µ −ν are
11
given in terms of the points zk and z at which they arrive on the on the asymptotic S 2
and their energies Ek , ω
Ek
(1 + zk z̄k , z̄k + zk , i(z̄k − zk ), 1 − zk z̄k ) ,
1 + zk z̄k
ω
qµ =
(1 + z z̄, z̄ + z, i(z̄ − z), 1 − z z̄) ,
1 + z z̄
1
−µ = √ (z, 1, i, −z).
2
pµk =
(6.7)
Rewriting the soft formula (6.5) in terms of the variables in (6.6) and (6.7), defining
(1)
(1)
Ŝzz ≡ ∂z X µ ∂z X ν Sµν , and acting with Dz2 gives [20]
Z
Z
κ
(1)
2
Im
duDz Cz̄z̄ − dvDz2 Cz̄z̄ =
[D2 Ŝ (1) − Dz2 Ŝz̄z̄ ].
8π z̄ zz
(6.8)
The formulae for the angular momentum and stress energy of a particle emerging at zk in
the appendix enables this to be rewritten:
Z
Z
2 zz
Im
duDz C − dvDz2 C zz
X
i
zk z̄k
= −8G
γ
Im Luz (zk )∂z̄k G(zk ; z) + hk ∂zk ∂z̄k G(zk ; z)
2
k
Z
Z
Z
1
2
duTuw − dvTvw .
= − Im d w∂w̄ G(w; z)
π
(6.9)
where total angular momentum conservation has been used. To summarize, the subleading soft graviton theorem [17], after some differentiation, change of notation and Fourier
transform, becomes the formula for the contour integrals characterizing spin memory.
7. An infinity of conserved charges
A conserved charge enables one to determine the outcome of a measurement on I +
from a measurement on I − . For example, for a process which begins and ends in a vacuum,
the total integrated outgoing energy flux across I + equals the incoming energy flux across
I − . In this section we describe an inf inite set of I + measurements – one for every contour
C – whose outcome is determined by a measurement on I − .
Conserved charges on I + may be obtained from any moment of the curl ∂[z̄ Nz] on
+
I−
. Consider for example
Q(z, z̄) = i∂[z̄ Nz] (z, z̄)|I + .
−
12
(7.1)
This can be written using the constraints as an integral over a null generator of I + .
Integrating by parts then gives the I + expression
Z
2 zz
1
2 z̄ z̄
Q(z, z̄) = −i du ∂z̄ ∂z Dz C − Dz̄ C
− ∂[z̄ Tz]u .
4
(7.2)
On the other hand using the continuity condition (2.13) and integrating by parts, one finds
the I − expression
Z
Q(z, z̄) = −i
2 zz
1
2 z̄ z̄
dv ∂z̄ ∂z Dz C − Dz̄ C
− ∂[z̄ Tz]v .
4
(7.3)
Equating (7.2) and (7.3) gives the conservation law:
Z
2 zz
1
2 z̄ z̄
du ∂z̄ ∂z Dz C − Dz̄ C
− ∂[z̄ Tz]u
4
Z
2 zz
1
2 z̄ z̄
= dv ∂z̄ ∂z Dz C − Dz̄ C
− ∂[z̄ Tz]v .
4
(7.4)
This conservation law equates the stress energy flux through and a zero mode of the metric
fluctuations along a null generator of I + to the same quantities on the P T -conjugate
generator of I − . There is one such law for every null generator.
The meaning of the conservation law (7.4) is a bit obscured by the fact that zero
modes of metric fluctuations are hard to measure. However, in the preceding we have
found that the time delay effectively measures a particular combination of the zero modes.
Our main formula (5.9) can be rephrased as a conservation laws for the charge
1
QC ≡ − 2 Im
π L
Z
Z
2
d wγww̄
d2 z∂z̄ G(z; w)Nz |I + ,
−
DC
(7.5)
which, as noted above, involves only the curl of Nz . Using (2.13) QC may be rewritten as
a I − charge
1
QC ≡ − 2 Im
π L
Z
Z
2
d wγww̄
DC
d2 z∂z̄ G(z; w)Nz |I − .
(7.6)
+
Integrating by parts, using the constraints to express (7.5) and (7.6) as integrals over I ± ,
and equating the two expressions yields
Z
Z
Z
1
+
2
2
∆ u + 2 Im
d wγww̄ d z∂z̄ G(z; w) duTuz
π L
DC
Z
Z
Z
1
−
2
2
= ∆ v + 2 Im
d wγww̄ d z∂z̄ G(z; w) dvTvz .
π L
DC
13
(7.7)
Thus if we measure the component Tvz of the radiative stress-energy flux and the
time delay on C at past null infinity, we can determine a moment of Tuz and the time
delay of an antipodally-located contour at future null infinity. There are infinitely many
such conservation laws – one for every contour C – which infinitely constrain the scattering
process.
Should it persist to the quantum theory, this infinity of conservation laws has considerable implications for the black hole information puzzle. The output of the black hole
evaporation process, as originally computed by Hawking, is constrained only by energymomentum, angular momentum and charge conservation. Imposing the infinity of conservation laws (7.7) (together with a second infinity arising from BMS invariance [24]) will
greatly constrain the outgoing Hawking radiation. These constraints follow solely from
low-energy symmetry considerations, and do not invoke any microphysics. It would be interesting to understand how the semiclassical computation of black hole evaporation must
be modified to remain consistent with these symmetries.
Acknowledgements
We are grateful to D. Christodoulou, S. Gralla, T. He, D. Kapec and P. Mitra for
useful conversations. This work was supported in part by NSF grant 1205550.
14
Appendix A. Massless particle stress-energy tensor
We start with the trajectory of a massless point particle:
pµ
xµ (τ ) =
τ + bµ
(A.1)
E
where pµ is the particle’s four momentum, with pµ pµ = 0. bµ = (0, bi ) = xµ (0) describes
the impact parameter of the straight-line trajectory relative to the spacetime origin. The
orbital angular momentum of this trajectory is:
Lµν = bµ pν − pµ bν ,
which implies
bµ =
1 µ0
L .
E
(A.2)
(A.3)
The total angular momentum is
J µν = Lµν + S µν
(A.4)
where Sµν is the intrinsic spin. The large τ behavior of the trajectory (A.1) is:
r(τ ) = τ +
1 µ
p Luµ + O(τ −1 )
2
E
1 µ
p Luµ + O(τ −1 )
E2
p1 + ip2
1 z
z(τ ) =
+
L + O(τ −2 )
E + p3
E u
u(τ ) = −
where we have used L0µ = Luµ and
1
z
E Lu
(A.5)
is O(τ −1 ). The matter stress-energy tensor of
the point particle is [28]:
Z
Z
δ 4 (y ρ − xρ (τ ))
δ 4 (y ρ − xρ (τ ))
M ρ
ρ
√
√
Tµν (y ) = E dτ ẋµ ẋν
−∇
dτ Sρ(µ ẋν)
.
−g
−g
(A.6)
Using Szz̄ = ir2 γzz̄ h near a particle with helicity h, a collection of point particles obeys:
X
δ 2 (z − zk )
M
lim r2 Tuu
=
Ek δ(u − uk )
r→∞
γzz̄
k
(A.7)
2
X
i
δ (z − zk )
2 M
lim r Tuz =
δ(u − uk ) Luz (zk ) − hk ∂z
r→∞
2
γzz̄
k
where uk ≡
− E12 pµk Lkuµ
k
and
1 ∂xµk ∂xνk
Lkµν
r→∞ r ∂uk ∂zk
b1 (1 − z̄k2 ) − ib2k (1 + z̄k2 ) − 2b3k z̄k
Ek .
= k
(1 + zk z̄k )2
Luz (zk ) ≡ lim
15
(A.8)
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